To determine how much government spending should increase to achieve full employment in a Keynesian model, we first need to calculate the initial equilibrium income using the given data.
Given Data:
- Autonomous Consumption (C₀): R100 million
- Investment (I): R200 million
- Government Spending (G): R300 million
- Marginal Propensity to Consume (MPC): 0.8
- Tax Rate (t): 0.25
- Full Employment Level of Income (Yₑ): R2,000 million
Step 1: Calculate the Initial Equilibrium Income
In a Keynesian model, the equilibrium income (Y) can be expressed as:
\[ Y = C + I + G \]
Where C (consumption) is influenced by income and the tax rate. The consumption function in this context takes taxes into account:
\[ C = C_0 + MPC \times (Y - T) \]
Where T (total taxes) can be calculated as:
\[ T = t \times Y = 0.25 \times Y \]
Substituting T in the consumption function:
\[ C = C_0 + MPC \times (Y - 0.25Y) = C_0 + MPC \times (0.75Y) \]
Now, substituting the values we have:
\[ C = 100 + 0.8 \times 0.75Y = 100 + 0.6Y \]
Step 2: Set Up the Equation for Y
Now we can set up the equilibrium equation:
\[ Y = C + I + G = (100 + 0.6Y) + 200 + 300 \]
This simplifies to:
\[ Y = 600 + 0.6Y \]
Step 3: Solve for Y
To isolate Y, we move the \(0.6Y\) to the left:
\[ Y - 0.6Y = 600 \] \[ 0.4Y = 600 \] \[ Y = \frac{600}{0.4} = 1500 \text{ million} \]
Thus, the initial equilibrium income is R1,500 million.
Step 4: Calculate the Change Needed to Reach Full Employment
To achieve full employment at Yₑ = R2,000 million, we need to find the required change in output (ΔY):
\[ \Delta Y = Y_e - Y = 2000 - 1500 = 500 \text{ million} \]
Step 5: Calculate the Multiplier Effect
The spending multiplier (k) is calculated as:
\[ k = \frac{1}{1 - MPC(1 - t)} = \frac{1}{1 - 0.8(1 - 0.25)} = \frac{1}{1 - 0.8 \times 0.75} = \frac{1}{1 - 0.6} = \frac{1}{0.4} = 2.5 \]
Step 6: Calculate the Required Increase in Government Spending
To find the necessary increase in government spending (ΔG) for the required increase in income (ΔY), we use:
\[ \Delta Y = k \times \Delta G \]
Rearranging gives:
\[ \Delta G = \frac{\Delta Y}{k} \] \[ \Delta G = \frac{500}{2.5} = 200 \text{ million} \]
Conclusion
To achieve full employment, government spending should increase by R200 million.